Given a sequence of random variables $\mathbf{x}=(X_{1},X_{2},\cdots ,X_{n},\cdots)$, with $X_{i}\in\mathsf{L}^{2}(\Omega,\mathcal{F},\mathrm{P})$ for each $i$ such that $\left\Vert X_{i}\right\Vert =\operatorname{E} (\left\vert X_{i}\right\vert ^{2})^{1/2}<+\infty$, $\sup(\left\Vert X_{1}\right\Vert ,\left\Vert X_{2}\right\Vert ,\cdots,\left\Vert X_{n}\right\Vert ,\cdots)<+\infty$, and $\inf(\left\Vert X_{1}\right\Vert ,\left\Vert X_{2}\right\Vert ,\cdots,\left\Vert X_{n}\right\Vert ,\cdots)=\sigma>0$.
Let $\mathbf{h}=(h_{1},h_{2},\cdots,h_{n},\cdots)\in \ell^{\infty}$ be a sequence of real number, such that $h_{i}\in\mathbb{R}$ for each $i$, and $\left\Vert \mathbf{h}\right\Vert _{\infty}=\sup(\left\vert h_{1}\right\vert ,\left\vert h_{2}\right\vert ,\cdots,\left\vert h_{n}\right\vert ,\cdots)<+\infty$. Define $\mathbf{h}^{\prime}\mathbf{x} =\sum_{i=1}^{\infty}h_{i}X_{i}$ and $$ \mathsf{S}=\left\{ \mathbf{h}^{\prime}\mathbf{x}:\mathbf{h}\in\ell^{\infty },\sum_{i=1}^{\infty}\operatorname{E}(\left\vert X_{i}\right\vert )<+\infty\right\} $$ Can I find some $p$ with $1<p<2$ such that $\mathsf{S}\subset\mathsf{L}^{p} (\Omega,\mathcal{F},\mathrm{P})$. One may tighten or relax certain conditions in the definition of $\mathsf{S}$.
Remark: In a financial market, $\mathbf{x}$ is the set of given risky assets (infinitely many), and $\mathsf{S}$ is the payoff space of the market. Since the market portfolio is $\mathbf{h}=\mathbf{1}=(1,1,1,\cdots)\in\ell^{\infty} $, hence we require that the holdings $\mathbf{h}\in\ell^{\infty}$, and a portfolio is an infinite linear combination.
For the market portfolio, $Y=\mathbf{1}^{\prime}\mathbf{x}\notin\mathsf{L} ^{2}$: let $\operatorname{E}(X_{i}X_{j})=0$ for $i\neq j$, then $$ \left\Vert Y\right\Vert ^2=\operatorname{E}\left( \left\vert \sum_{i=1} ^{\infty}X_{i}\right\vert ^{2}\right) =\sum_{i=1}^{\infty }\operatorname{E}\left( \left\vert X_{i}\right\vert ^{2}\right) \geq\sum_{i=1}^{\infty}\sigma^2=+\infty $$
For any portfolio $Y=\mathbf{h}^{\prime}\mathbf{x}\in\mathsf{S}$, there is $\operatorname{E}(\left\vert Y\right\vert )<+\infty$: by $$ \left\vert Y\right\vert \leq\sum_{i=1}^{\infty}\left\vert h_{i}X_{i} \right\vert \leq\sum_{i=1}^{\infty}\left\vert h_{i}\right\vert \left\vert X_{i}\right\vert \leq\sum_{i=1}^{\infty}\left\Vert \mathbf{h}\right\Vert _{\infty}\left\vert X_{i}\right\vert =\left\Vert \mathbf{h}\right\Vert _{\infty}\sum_{i=1}^{\infty}\left\vert X_{i}\right\vert $$ we have $\operatorname{E}(\left\vert Y\right\vert )\leq\left\Vert \mathbf{h}\right\Vert _{\infty}\sum_{i=1}^{\infty}\operatorname{E}(\left\vert X_{i}\right\vert )<+\infty$.